an analysis of parameters for the johnson-cook strength ...subscript i represents an arbitrary...
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Army Research Laboratory Aberdeen Proving Ground, MD 21005-5066
GRL-TR-2528 June 2001
. An Analysis of Parameters for the Johnson-Cook Strength Model for 2-in-Thick Rolled Homogeneous Armor
Hubert W. Meyer, Jr. and David S. Kleponis Weapons and Materials Research Directorate, ARL
Approved for public release; distribution is unlimited.
Abstract
Yield strength obtained from quasi-static strength data for rolled homogeneous armor @HA) was combined with dynamic strength data for 2-in (5lmtn) RHA to generate Johnson-Cook parameters for 2-in RHA. One parameter was fixed based on the quasi- static strength data, and a least-squares method was used to fit the others individually. The fit was tested with CTH by simulating the penetration of stacks of 2.5~in-thick (63.5-mm) RHA plates (the closest available experimental data). Parameter analysis and comparison of the simulations to experiment substantiated the approach.
.
ii
Acknowledgments
The authors would like to acknowledge Shuh Rong Chen for providing the raw data (in
digital form) that was used in this report.
This work was supported in part by a grant of high-performance computing time from the
Department of Defense High Performance Computing Center at Aberdeen Proving Ground, MD.
.
Table of Contents
.
. . . Acknowledgments . . . . . ..*......*.~...*...*.*.**.. ..~.....,..~...,~....~.....~,........~...~.....................~.~.. 111
List of Figures . . . . . . . . . . . . . . . ..*.*.*.......**........ . . . . . . . ..*......*.......*.......*...............*................... vii
List of Tables . . . . . ..*.~I..............~**~~~~..~~~~~....**......... . ..**~...*...~+~I~*.~..*~......,..~......*.......*.*.. vii
2. Dynamic Data . . . . . . . . . . . . . ..~....~~...................~........*.......~.................................*...*.*.*~.*~*. 3
3. Quasi-Static Data .*.*.*.~....*~.*....*...****..*.. . . . . . ..~.*.~.....**~*.~.*..*..*~**.~.*.~.....~.~~......~*~~-.--..~.. 4
4. Fitting the Parameters ............................................................................................. 5
4.1 Parameter A ......................................................................................................... 5 4.2 Parameters B and n .............................................................................................. 6 4.3 Parameter c .......................................................................................................... 8 4.4 Parameter m ......................................................................................................... 10
5. Numerical Simulations ............................................................................................ 13
5.1 Setup .................................................................................................................... 13
5.2 Results .................................................................................................................. 14
5.3 Discussion ............................................................................................................ 17
Distribution List .~.~..~.............,...~...~*.....**~****..***.*.*~*.*~********.~~*.*..*......~.*.~~~..~. . ...*.....*.. 23
Report Documentation Page . . . . ..***..................... ..~~........*.**..*.~.~....................****...*** 25
v
List of Figures
Figure m
1. RHA Hardness Variations Specified by MIL-A- 1256OH *.*.**.*..e....*..*............*...*...... 1
2. Dynamic Strength of 2-in RHA . ..~....****.*.......~.*....**...~~...~.......................*.....~....~....~.* 4
3. Quasi-Static Yield Strength of RHA.. ........................................................................ 6
4. Simulation Setup.. ...................................................................................................... 14
5. Nose and Tail Tracer Histories for the 1,616 m/s, Set 2 Case ................................... 16
6. Set 2: 1,616-m/s Penetration Plot ............................................................................. 18
List of Tables
Table Pap;e
1. Dynamic Strength Data for 2-in RHA ....................................................................... 4
2. Quasi-Static Yield Strength of RHA .......................................................................... 5
3. Strain-Rate Dependence of Room-Temperature Data ............................................... 9
4. Quadratic Fit of the Temperature Data ...................................................................... 12
5. RHA Parameter Sets Evaluated ................................................................................. 1s
6. KHA Penetrations in Millimeters for the Parameter Sets Evaluated ......................... 15
vii
1. Introduction
Class 1 rolled homogeneous armor @HA), designed for maximum penetration resistance, is
available in thicknesses from l/4 in (6.35 mm) up to 6 in (152.4 mm). Military specification
ME-A-12560H (U.S. Department of Defense 1991) allows a wide variation in hardness over the
range of available thicktresses as well as within each thickness group (Figure 1). Since hardness
is an indicator of several material strength properties, a significant variation in material
properties exists over the range of thicknesses of available FWA and to a lesser extent, within
each thickness group.
400
375
325
275
250
225 0 1 2 3 4 5 6
Nominal Thickness of Plate, in
Figure 1. RHA Hardness Variations Specified by MIL-A-12560H.
To properly model the ballistic performance of FWA, consideration must be given to these
property variations. The wide variation in material properties across the spectrum of available
thicknesses suggests that each thickness should be separately evaluated to obtain valid strength
parameters. Further complications exist (e.g., manufacturing lots and through-the-thickness
hardness variations). However, these complexities are avoided in the present work by assuming
1
that the variations in properties for a particular thichess, as allowed by the thickness group, are
negligible. That is, for an RHA plate that conforms to MIL-A-12560H, specifying its thickness
is sufficient in identifying its properties.
The shock physics code CTH (McGlaun et al. 1990) is used at the U.S. Army Research
Laboratory (ARL) to model ballistic impact and penetration experiments. The Johnson-Cook
strength model (Johnson and Cook 1983) is one of several strength models available in CTH. It
is an empirical model that computes material flow stress as a function of strain (work) hardening,
strain-rate hardening, and thermal softening. The Johnson-Cook model takes the following form:
Y=A I+;$ (l+Cln$*)(I-T*-), ( 1
where A, B, C, m, and n are constants, E is the equivalent plastic strain, ti* is the strain rate
nondimensionalized by the reference strain rate of l/s, and T* is the nondimensional
temperature. Parameter A, the initial (E = 0) yield strength of the material at a plastic strain rate
of k = l/s and room temperature (298 K), is modified by a strain-hardening factor (containing
parameters B and n), a strain-rate-hardening factor (containing parameter C), and a
thermal-softening factor (containing parameter m).
T’ is defined by
-f = T-T,
T, -T, ’ (2)
where T, is room temperature and T,,, is the melting temperature of the material, 1,783 K for
RHA. Equation (2) is the form used in CTH and is valid for Tr I T I Tm, the region of interest in
most ballistic applications,
2
CTH originally contained a single set of parameters that had been typically used in
simulations for any thickness of FW.A. These parameters were taken from one of two data fits
for RHA presented in Gray et al. (1994). Both of these fits (which will be discussed) were
determined using 2-in-thick FW4 that conformed to ML-A-12560H. The fits resulted in
overprediction of the quasi-static yield strength (A in equation [Xl). Their approach to
optimization was to consider all parameters simultaneously. This approach to fitting the data
resulted in a model for the RHA that under-predicted the depth of penetration of several
experiments; this is discussed in more detail later. In the present work, Johnson-Cook
parameters are developed for that particular batch of 2-in-thick RHA. The approach taken here
is to fix the value of A based on the quasi-static test data. Au optimum fit to the data for each of
the remaining parameters is then found individually, as suggested by Johnson and Cook (1983).
2. Dynamic Data
Gray et al. (1994) generated compressive stress-strain data for a variety of metals over a
range of temperatures and strain rates using the split-Hopkinson pressure bar. The digital data
consisted of the results of dynamic tests of 2-in-thick RHA. At room temperature, tests were
conducted at four strain rates (0.001; 0.1; 3,500; and 7,000/s). At elevated temperatures (473 and
673 K) tests were conducted at a strain rate of 3,000/s. Strains were recorded from near zero up
to about 0.20.
l
To expedite processing time and utilize all of the available data, the digital data was not used
directly to obtain the Johnson-Cook parameters, rather it was fit to analytical functions that were
suitable to the software available for use during this study. The fits of the six data sets are shown
.
graphically in Figure 2 and algebraically in Table 1, The functions in Table 1 are fits to the RHA
strength data from Gray et al. (1994) and are used to determine the Johnson-Cook parameters in
the following analyses. For clarity, yield strength predicted by the Johnson-Cook model is
denoted by Y (in GPa), whereas y (in GPa) represents the data fits.
T=298K i=7,OCO/s
T=298 K, i=3,soo Is
T=298 K, 1=0.100 IS T=298Y .k=O.OOl/s
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
True Strain
Figure 2. Dynamic Strength of 2-in RHA.
Table 1. Dynamic Strength Data for 2-in RHA
II Temperature Strain Rate Function Equation No. K) Us) II
I n nnl ’ -. 1.4905E oMo7 1 (3a) II
I -
298 1 ,= 298 3,5oc 298 473 673
b 7,000 3,000 3.000
_I 1.5206~ o.w23 v = 1.5935E o.0529 y = 1.6048~ o*0415 y = 1.3410E o’0231
i mm o.0357
(W (3c) (34 (W (?fi
3. Quasi-Static Data
T=473K i=3.OoO/s
f
T=673K, i=3,OOO/s I
Benck (1976) determined several properties for three thicknesses of RHA. He measured the
quasi-static tensile yield strength in the three principal plate directions (in the rolling direction,
across the rolling direction, and through the thickness) at a strain rate of 0.0003/s. For present
purposes, these values were averaged to obtain a representative isotropic value. The
I 4
compressive yield strength of the material is then assumed to be equal to the tensile yield
strength; this is only approximately true for RHA. The data are presented in Table 2 and include
unpublished data for 3/16-in (4.76 mm) RHA (Bruchey 1997).
The values from Table 2 and an analytical fit to these data are plotted in Figure 3. A
logarithmic form was chosen; the computed fit of the data is
y = (- 0.142X@ + 0.8772, (4)
where y is the yield strength in GPa and t is the plate thickness in inches.
Table 2. Quasi-Static Yield Strength of RHA
Plate Thickness Yield Strength (in b4) @Pa>
0.1875 [4.76] 1.14 0.5 [ 12.71 0.94 1.5 [38.1] 0.82 4.0 [101.6] 0.69
4. Fitting the Parameters
4.1 Parameter k Parameter A is the yield strength at room temperature and a strain rate of
l/s. Equations (3a) through (3d) in Table 1 were interpolated to generate a function describing
the behavior of the 2-in RHA at a strain rate of l/s. The resulting function is
y = 1.5384s o.0436. (5)
A comparison of equations (5) and (3a) (Table 1) shows a difference of less than 2% between the
yield strength at f; = l/s and i: = 0.001/s at a strain of 0.01. Furthermore, Benck and Robitaille
(1977) report a difference of about 1.1% for 38-mm RHA plate and about 0.6% for loo-mm
5
0 0.5 1 1.5 2 2.5 3 3.5 4
Thickness of RHA, inch
Figure 3. Quasi-Static Yield Strength of RHA.
RHA plate between the quasi-static yield strength at 0.0003/s and at 0.42/s. For the present
work, the value of A is approximated by the quasi-static data (Table 2 and equation [4]). A value
of A = 0.78 GPa for 2-in-thick RHA is obtained from equation (4).
4.2 Parameters B and n. The remaining parameters from equation (1) (B, C, m, and n) are
frt to the functions bf Table 1 by a least-squares technique. To fit the parameters B and n, write
the first two terms of equation (1) as
(6)
Equation (6) represents the yield strength at room temperature and strain rate of l/s, conditions
that render the last two terms in equation (1) equal to unity. Equation (6) is rearranged to
6
Y -A=Bs”. (7)
Let
g=ln(Y-A) (8)
so that
q=nlns+b, (9
where b=lnB.
The data at these conditions are given by equation (5), which is of the form
y=pEa. (10)
The data must be in the same form as equation (8), so A is subtracted from both sides of
equation (lo), leading to the following representation of the data:
Y=ln(y-A), (11)
and
‘3 =ln@~” -A). (12)
The error incurred by approximating the data (equation [ 111) with the model (equation [8 3) at a
strain &i is ‘pi - Yi. Subscript i represents an arbitrary discretization of the data into seven
strains covering the range of the data (E = 0.01, 0.02, 0.04, 0.08, 0.12,0.16, and 0.20). This was
done to simplify the fitting procedure. In the least-squares method, the error is squared (to avoid
having positive and negative errors combining arithmetically to reduce the total error) and
summed over the range of the data; the sum of the squared errors is to be minimized:
7
I 7 C( Cj3pi -Yi)2 =minimum i=l
(13)
The sum can be minimized with respect to parameters I3 and n if the derivatives are set equal to
zero. That is,
$~( Ipi -Yi)2 =O, 1-I
and
(14)
(15)
Equation (9) is substituted, and the differentiation is carried out, resulting in the following two
equations:
(cln&i)z -7C(lnEi)2 ’
and
n=CYi-7b ChE, ’
(16)
(17)
where Yi = Yi (ei ) is known (equation [ 12]), and the summation indexes have been omitted for
clarity. The results are I3 = 0.78 GPa and n = 0,106. These results minimize the error; this was
verified by det ermining that the derivatives of equations (14) and (15) were positive.
4.3 Parameter C. The first two factors in equation (1) are
8
Y(s,,L* -1, T’=O)=A =Si, (18)
where, for simplicity, this contribution to the strength is termed Si, Thus, for room temperature,
equation (1) becomes
Yi =si(l+chI~*)(l-o)=si +sichG*. (19)
To obtain a corresponding expression for the data, equations (3a) through (3d) (Table 1) are
used to generate curves of stress vs. h~i’ for various constant strains. To generate the curves, the
first of the seven discrete strains was substituted into each of equations (3a) through (3d)
(Table 1) to generate stresses for each of the four strain rates. The resulting stress was plotted
against
awes.
hri?’ , and the process repeated for each of the remaining six strains, resulting in seven
Analytical expressions were fit to the curves; the results are detailed in Table 3.
Table 3. Strain-Rate Dependence of Room-Temperature Data
i so y=ui+Vi~~* Equation No. Ui Vi
1 0.01 1.2587 0.0035 (2Oa)
I 2 3 0.04 0.02 1.2972 1.3370 0.0039 0.0044 CW (2Oc) I 4 0.08 1.3780 0.0050 (20d) 5 0.12 1.4026 0.0053 (20e) 6 0.16 1.4203 0.0055 (2Of) 7 0.20 1.4342 0.0057 (2On)
(20)
These data can be represented by the form
yi =Si +Siki~*, (21)